Consider the following regression model: Yi = αXi + Ui , i = 1, .., n...

Consider the following regression model: Yi = αXi + Ui , i = 1, .., n (2)

The error terms Ui are independently and identically distributed with E[Ui |X] = 0 and V[Ui |X] = σ^2 .

1. Write down the objective function of the method of least squares.

2. Write down the first order condition and derive the OLS estimator αˆ.

Suppose model (2) is estimated, although the (true) population regression model corresponds to: Yi = β0 + β1Xi + Ui , i = 1, .., n with β0 different to 0.

3. Derive the expectation of αˆ, E[ˆα], as a function of β0, β1 and Xi . Is αˆ an unbiased estimator for β1? [Hint: Derive first E[ˆα|X].]

4. Derive the conditional variance of αˆ, V[ˆα|X], as a function of σ^2 and Xi .

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Expert Solution

milcah answered 3 months ago

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